The notation used here is different from that used in [1]}. Indeed, the variable \(g\)  that labels the ket \({g}\)  has a slightly different meaning. For us, \({g}^A_B\)  must become \({g}^B_A\)  because \(g\)  is a fact shared symmetrically by \(A\)  and \(B\) . In [1]} instead, \({g}^A_B\)  must correspond to \({g^{-1}}^B_A\)  because \(g\)  labels a transformation from \(A\)  to \(B\) . The relation between the two notations is given by the maps \(\mathcal {I}^A_B\)  which turn facts into transformations.
